Step 1 :Given the function \(f(x)=8x^{2}-9\), where \(x \geq 0\), we are asked to find its inverse.
Step 2 :First, we replace \(f(x)\) with \(y\), so the equation becomes \(y=8x^{2}-9\).
Step 3 :Next, we swap \(x\) and \(y\) to get \(x=8y^{2}-9\).
Step 4 :Solving for \(y\), we get \(y^{2}=(x+9)/8\).
Step 5 :Since \(x \geq 0\) in the original function, we only take the positive square root, so \(y=\sqrt{(x+9)/8}\).
Step 6 :Finally, replacing \(y\) with \(f^{-1}(x)\), we get \(f^{-1}(x)=\sqrt{(x+9)/8}\), which is the inverse of the function \(f(x)\).
Step 7 :So, the correct answer is \(\boxed{f^{-1}(x)=\sqrt{(x+9)/8}, x \geq-9}\).